I don’t think it an overstatement to say that the concept of the infinite plays a key role in the philosophy of religion. There are at least two senses in which ‘infinite’ is used. First, ‘infinite’ is often used to mean maximal, as in God’s infinite power, knowledge, and goodness. Second, many arguments in the philosophy of religion discuss ‘infinite number’ or ‘infinitely many’. It is this second sense of the infinite that I focus on in this post. Here are two recent examples of this second sense of the infinite, from Prosblogion, with select quotes (and links to the full posts):

Grim Reapers vs. Uncaused Beginnings
By Joshua Rasmussen on February 4, 2013 10:47 AM
‘Another idea is that there is a problem with producing an actual infinite number of events. This third idea, if correct, would seem to block the grim reaper argument against uncaused beginnings. But it would reinforce the argument for a finite past.’Infinite multiverse, fine-tuning and probability
By Alexander Pruss on March 6, 2013
‘So it seems that the only reasonable place to put the probability shift is when you find out that there are infinitely many Joneses who rolled a die.’
In these posts, we have two puzzles, one involves what happens if there are countably many grim reapers, the seconds puzzle involves probabilistic reasoning if there are countably many Joneses. Indeed, puzzles and paradoxes of the infinite are many. Yet few have stopped to ask the question: Which objects are the infinite natural numbers? Put another way: How should the finite natural numbers, numbers like 7 and 113, be extended into the infinite? Put still a third way: When someone says ‘there are an infinite number of grim reapers’ or ‘there are infinitely many Joneses’, what sort of structures should the person be referring to?
I simply assume that there are better and worse ways for concepts to carve up the world. I also assume that the more something walks and quacks like a duck, the more likely it is to be a duck. Then, infinite natural numbers in a nonstandard model of the reals behave very much like finite natural numbers, are so are the correct extension of the finite natural numbers into the infinite. When someone says ‘there is an infinite past consisting of an infinite number of days’, the days should be the structure of an infinite natural number. This blocks any sort of Grim Reaper problem. If there are infinitely many Joneses, this must be referring to an infinite number (in a nonstandard model) of Joneses, and then the reasoning becomes analogous to the finite case. Indeed, I suggest that correctly using ‘infinite number’ and ‘infinitely many’ blocks all paradoxes of the infinite. For more, see the papers here and here, and a video here.
It might be asked: ‘But why can’t we discuss days of the structure of omega-star, that is: …-3, -2, -1, 0? Surely this is an example of an infinite number!’ No, it isn’t. Certainly it is infinite, but it is not an infinite number. And, as Aristotle held, such an infinity is always potential, never determined and actual. It is inexhaustible. If there is an infinite past, it is the structure of an infinite natural number (which, it should be noted, has a beginning). Infinite natural numbers in a nonstandard model of the reals are actual and determined. For this distinction, as well as a test to determine whether something is a potential or actual infinity, see here.
If infinite numbers are going to play a key role in reasoning, it might be a good idea to first figure out what the infinite numbers are.

I’m all for studying alternative mathematical theories (e.g. non-well-founded set theories, alternative theories of the continuum, etc.). But, at the risk of seeming to be deaf, I don’t understand your concern. You’ve pointed to no problem whatsoever in the standard account of infinite number. What is the problem you’re trying to solve? And what is the cost of your solution? (Or, as Lewis once said: YOU go tell the mathematicians that they’re wrong…).

Jeremy:
I don’t think anything in these arguments depends on the concept of infinite number. Rather, the arguments depend on the concept of a countably infinite plurality, i.e., a plurality such that the set of the things in the plurality will be a countable infinity. So, really, it is the concept of a countably infinite set rather than an infinite number that is involved.

Eric: My concern is: What are the infinite numbers? This concern, largely, stems out of the host of puzzles and paradoxes surrounding the infinite. Most mathematicians donât worry about the infinite, and run into no problems because they always treat the Cantorian infinite (e.g., omega) properly â namely, as a potential infinity, one that is not actual and determined. As a simple example, consider an infinite sum, or calculus as traditionally taught/done.
It is, to a great extent, philosophers who come along and talk about infinitely many Joneses, or an infinite number of grim reapers, or a sphere with an infinite number of shells (a Zeno-sphere). That is, it is largely philosophers who treat omega as infinite number, and as a completed, actual thing. Then problems ensue (if you disagree, read the torturous literature on what happens when two Zeno-spheres collide). I also think mathematicians run into trouble in some cases, for example, with the Continuum Hypothesis. There also are mathematicians who do not drink the Cantorian Kool-aide, e.g., here.
If one believes that, in part, philosophy is about clarity and conceptual analysis, it would seem important to figure out what the infinite numbers are. I donât quite see how this can be denied.
Alex: But whether a countable infinity can be treated as actual and determined is a key question. And in large part, I think people want to treat countable infinities as actual and completed because they think that they are dealing with the infinite numbers (which, I suggest, they are not).

Jeremy:
I believe that after every year of my future, there will be another year, and there is no circularity in time. Eternalism is also true. So consider my future years! Voila, a completed (and well-ordered) actually infinite plurality of years.
But perhaps it’s uncountable. So, now, use mathematical induction to generate a function from the natural numbers to the years of my future, by using the rules: f(0) = AD 2014 and f(n+1) = one year after f(n). Those years that are in the range of f are a completed actual countable infinite plurality of years.
I fail to see where I acted as if I was dealing with infinite numbers.

Alex: Your argument seems the same as assuming that Platonism is true and considering the natural numbers, in order to arrive at a completed actual countable infinity. Perhaps not surprisingly, I am not swayed by this argument. If you assume that the natural numbers are a completed actual entity, it is not hard to prove that the natural numbers are a completed actual entity.
Perhaps though, there is more to your argument so that is not analogous to my recreation?

1. I use an inductive definition. If the use of inductive definitions commits one to Platonism, then all serious mathematics commits one to Platonism.
2. I don’t even think the second part of the argument is needed, because I think that the infinity of my future years is countable. I added that just in case I’m wrong and the afterlife is uncountable.

Donât you think that Eternalism is smuggling in Platonism?
Or, to put the point differently, if the argument âEternalism is true, consider future days, voilaâ shows that the natural numbers are unique, determined, and actual, why have so many people been confused for so long into thinking that the natural numbers (if such a thing exists uniquely) are a potential infinity? Are there really no weak points in this argument? I ask this seriouslyâ¦ as on a quick thought, it seems to me that Eternalism is smuggling in Platonism, or if not, is this really a knock down argument?
Also, I perhaps should reiterate that, regardless of the status of the natural numbers, I still think that language matters, and that what we call things matters; it is important to determine which objects the infinite natural numbers are. Though indeed, if the natural numbers are unique, actual, and determined, then it would be possible to speak of them as suchâ¦âconsider an actual, determined countable infinityââ¦. (But, it seems to me, there is overwhelming evidence that they are not.)

Jeremy:
Certainly, eternalism is controversial. But it’s quite different from Platonism. I am not a Platonist–I am a theistic conceptualist about universals and a structuralist about mathematics. If I were an atheist, I’d be a nominalist, but I’d still be an eternalist.
I wasn’t giving a knock-down argument. I was just giving a reason that appears to be quite independent of thinking of infinity as a number.

I’m a little puzzled by the term infinite natural number you repeatedly use. Isn’t it fairly trivial that every member of the natural numbers is finite?
Also, I’m a little puzzled as to why you are treating omega as the standard bearer of Aristotles potential infinite – omega is defined as the order type of an actually infinite set.

Alex: Interesting. I get that sense that you think that the the natural numbers may not be determined and actual due to worries like the Grim Reaper. If this is correct, what do you think potential flaws are in your argument (Eternalism, my future days, voila)?
Kevin: I actually think that the finite/infinite divide is problematic, and that the finite natural numbers gradually become infinite. But that isn’t crucial to my point. I am talking about what also go by the name hyperintegers, see here:http://en.wikipedia.org/wiki/Hyperinteger
Abraham Robinson, and others, from time to time have called them infinite natural numbers. As for omega, I am being a bit fast and loose. I just mean the sort of infinity that begins somewhere and goes on. If discrete, you get something omega-like (thought maybe not in all detail), in the continuous case you get a ray.

Jeremy:
Grim Reaper makes me think that nothing can have an infinite number of things in its causal history. But this says nothing about simultaneous or future actual infinities.
That said, I am worried about how well-defined the natural numbers are because of upward Lowenheim-Skolem. I am much more confident of an actual infinity of future days than of the claim that that infinity is “really countable”. Hence the second part of the argument where I use induction to extract a countable subset of that infinity. Because of upward L-S I worry about that part of the argument.
But I really haven’t thought much about this.

Here is a bit of a challenge — (one that may prove easier than I imagine). Present a puzzle or paradox involving the infinite, but where ‘infinitely many’ or ‘an infinite number’ refers to infinite natural numbers. (At the least, such numbers have a certain structure, including first and last elements, which blocks many paradoxes and puzzles of the Cantorian infinite.)
Alex: But what about a spatial and future Grim Reaper problem — that is, there are Grim Reapers at distance 1, 1/2, 1/4, 1/8… They kill you if you reach them, starting at 0 and moving to the ‘right’. Isn’t this a problem of a future actual infinity? (I thought this sort of thing was Bernadete’s original example.)
Interesting — I too am concerned about how well-defined the natural numbers are. I suspect not very well. But I look forward to thinking more about your specific worries.

Jeremy:
These Grim Reapers are all going to be in a single backwards causal cone centered on some future point. And that’s what I object to: infinitely many things in the causal past of a single event.
As for the challenge, how about this paradox. For any infinite hyperinteger N (in a closed real field), the set { 0,2,…,N } is uncountable. But it is paradoxical to suppose an actual uncountable infinite that has no countable infinite subset.

Jeremy â
You wrote âMost mathematicians donât worry about the infinite, and run into no problems because they always treat the Cantorian infinite (e.g., omega) properly â namely, as a potential infinity, one that is not actual and determined. As a simple example, consider an infinite sum, or calculus as traditionally taught/done.â
Huh? I know lots of mathematicians who surely would not agree. And, philosophically, I would be in very good company to point out that Aristotleâs potential infinity / actual infinity contrast is widely thought to be incoherent. Whatâs âpotentialâ or not âactual and determinedâ about an infinite sum? Having written omegas over my sigmas many times, and seen the same in math book after math book, I donât grasp your point at all. (And yes, I understand epsilon-delta limits, and why you need the âactualâ infinity of the continuum to even make sense of them; e.g. I understand Dedekind.)
Your point about the applicability of the mathematics of infinity to our universe is quite good. But thatâs not a problem about math â it may well be that our universe does not have certain types of complexity. Fine. That wouldnât invalidate Cantor (or any other math). And the debates about Zeno spheres are fascinating debates that may indeed lead to novel insights into physicality and infinity. Fascinating work has been done on the types of continua needed to support hypertasks â uncountably long sequences of operations in finitely measurable space-times. Serious, deep, and fascinating work.
The Continuum Hypothesis is surely not a problem about infinite numbers. Itâs a problem about motivating axioms. After all, one can settle the Continuum Hypothesis with a single statement: the cardinality of the continuum is aleph-1. Done! Now you just work in ZFC+CH. The problem is with the axioms of ZFC, which, by the way, leave lots of open problems, not just the CH. When mathematicians propose, say, an axiom of determinacy, it isnât because they are trying to solve some mystery about infinite numbers. And when Woodin is working on the CH, he isnât worried about that âmysteryâ either. Heâs after much, much bigger fish. And heâs doing brilliant, brilliant work.
Everybody (including me) will agree that we need better axioms; but not because of vague Aristotelian mysteries about infinite numbers. And I think many mathematicians will also agree that we need better understandings of number systems in general â such as some of the deep algebraic issues surrounding Conwayâs systems. None of this has anything to do with dear old Aristotle.
Sure, a small minority of mathematicians havenât drunk the Cantorian kool-aide. But thatâs not an argument. An argument is that youâve got some better math for us. If you do, and it comes at a lower cost, then certainly, youâre on the right track. But so far you havenât shown that you do have better math. Do you have some axioms? Can they solve outstanding problems? What new theorems can you prove? Are your axioms consistent with ZFC? Are you really describing a new structure or just an old structure in new clothes? If new, prove it. Show us the math. More power to you if you can.
– Eric

With Eric, I don’t see how typical mathematicians who work with infinities treat the infinite as potential. In my own fields of mathematics–analysis and probability–we certainly treat real numbers as completed entities. We quantify over them, we attribute properties to them, we stick them into sets, and so on. Yet a real number is something like an equivalence class of infinite sequences of rationals or the like. We talk of measures on infinite structures all the time, and the measure is typically a function from an infinite set of subsets. Those subsets seem to be fully determined. Some of them are uncountable and others are countable. We do all sorts of things with them.
In analysis, we go multiple levels of abstraction from an infinity. A real number is an equivalence class of some infinite entities (sequences or cuts, say). A typical function (in analysis and probability) is an infinite set of pairs of real numbers. And then we talk of function-spaces like L^p which are infinite sets of functions. We may interpolate between these function-spaces, embed one in another, and so on. Maybe there is a way of making all of this consistent with seeing infinity as potential. But I don’t think that the typical mathematician in analysis thinks of it like that. In analysis, we tend to work in pictures. I have a mental picture of a function on some bounded infinite set. That picture doesn’t have gaps in it.

Alex: âThese Grim Reapers are all going to be in a single backwards causal cone centered on some future point.â I donât quite follow; arenât there just countably many guys ready to kill the poor walker? Could you explain the causal part a bit more.
That doesnât seem like much of a paradox to me. Iâm inclined to say that there can be a countable subset, as long as it is properly understood: It is not actual and determined. Though I should add, I donât think much about the uncountable, since I think that problems arise at the level of countable.
Alex and Eric: I would be curious what your response is to this paper:http://philpapers.org/rec/GWITDT
In it, I propose a mathematical criterion that indicates whether a structure is merely potentially infinite.
The reals certainly have constructive definition.
Infinite sums are merely potential, for example, the sum from 0 to infinity of 1/2^I never âreachesâ 2. A hyperfinite sum, of 0 to N, âreally getsâ infinitesimally close to 2. Though your points about much of classical mathematics are well taken. I suppose here I reply along the lines of JP Mayberry in The Foundations of Mathematics in the Theory of Sets. If I recollect, the thought is that not enough work has been done with different assumptions to tell which is the most fruitful way to do mathematics.
Eric:
âAnd the debates about Zeno spheres are fascinating debates that may indeed lead to novel insights into physicality and infinity.â
Sure â maybe you could explain to me, in a simple and coherent manner, what happens when 2 Zeno-spheres collide?
âThe Continuum Hypothesis is surely not a problem about infinite numbers.â
Yah, it is.
âAnd when Woodin is working on the CH, he isnât worried about that âmysteryâ either. Heâs after much, much bigger fish. And heâs doing brilliant, brilliant work.â
I deny none of this. I just worry that in 100 years it might be seen as misguided and largely pointless work.
I think what you are missing Eric, is that my main point is a very simple one about how our concepts should carve up the world. How should the finite whole numbers be extended into the infinite? All the questions I asked in my original post, actually. How anyone can deny that the infinite natural numbers are the correct extension is beyond me. Your questions at the end of your post about importance are good questions. At the least, I believe that carving up the world properly, properly referring to the infinite numbers, and understanding that certain infinities are merely potential dissolves paradoxes of the infinite, and explains why there might be trouble like CH.

By the way, the dart puzzle is yet another nonconglomerability puzzle, I think. The issue is that the event {r1 < r2} is not measurable on the relevant probability space, but once we have the value of r1, say, that it’s x1, then {x1 < r2} is measurable.

Jeremy:
Some more comments about your paper.
1. The puzzle is Sierpinski’s counterexample to the Fubini property, isn’t it?
2. You end by saying: “As argued above, this leads to the conclusion that two observers ï¬nd that r1<r2 and r2<r1.” That sounds paradoxical, but the paradox is decreased when we realize that the two observers have different information. The first observer knows the value of r1. The second observer knows the value of r2. Different conclusions are appropriate on the basis of different information. 🙂
3. That said, there is obviously something uncomfortable about the case. For one, it probably provides yet another counterexample to van Fraassen’s Reflection Principle. For if you’re the first observer, you know ahead of time that you will come to be almost sure that r1<r2 and if you’re the second observer, you know ahead of time that you will come to be almost sure that r2<r1. But you shouldn’t ahead of time think r1<r2 even if you’re the first observer, since surely the order of observation shouldn’t matter.
But this is because this is a nonconglomerability case. Nonconglomerability and violation of Reflection go hand-in-hand.
I am not trivializing this, but just putting it in the context of all the cases of nonconglomerability.

Alex: I look forward to reading up on those topics. I am curious, is there a standard response to such worries. The two observers’ situations do seem uncomfortable, to say the least.
Eric: And I am still waiting for your Zeno sphere explanation. You also seem to be entirely missing the purpose of this work. But here is a new claim, the natural numbers are too large to be a finite number and too small to be an infinite number. Follows on the correct view of infinite number.

Jeremy –
I have nothing against your project. I am asking to see it – to see your theory of numbers.
Give me some definitions and axioms. Prove some theorems, answer some open questions.
Far from being opposed to this sort of thing, I’m all in favor of it.
After all, I love math very much, so if you’ve got some better math, I want to learn about it.
– Eric

As I think that concepts are important, including figuring out what the infinite numbers are, I think the following is important. (I realize that it is unlikely to mean much to Cantorians intent on ignoring the question: Which objects are the infinite natural numbers?)
Definition: A Number, n, is an object of the form n = 0, 1, 2â¦, n-2, n-1.
Discussion: Order is involved. This definition is sensible, in that it accords with our conception of finite numbers, though the definition is not limited to finite numbers. The definition obviously differs from Cantorâs infinite ordinals for infinite n.
Assumption: For any two sets A and B, if A is a proper subset B, then A is smaller than B.
Discussion: This is (one version of) a Euclidean notion of relative size, in which the (proper) part, A, is strictly smaller than the whole, B. For good overviews of this position on relative size, see Mancosu (2009) and Parker. Note that though this conception of size is non-Cantorian and currently out of favor, it is perfectly consistent.
Theorem: The natural numbers are too large to be a finite number and too small to be an infinite number.

I am familiar with non-standard models of arithmetic (i.e. of the Peano Axioms). The only models considered here are countable.
I note that these non-standard models can be easily encoded in the standard models of the Peano Axioms. (That is, you can easily generate a structure over the standard natural numbers that is isomorphic to the nonstandard model.) Thus these nonstandard models are easily handled by Cantorian set theorists. The order relation in the nonstandard models are provably Turing-computable while the addition and multiplication relations are not (Tennenbaum’s theorem). Thus these nonstandard models are of little use or interest arithmetically and hardly have much claim to be either correct or superior to the standard models. Still, there is interesting structure to be studied here, precisely because of the non-Turing computability.
Having read your papers, what you are doing looks much like the non-standard models of Peano arithmetic, but not quite. Thus I would ask you to give precise axioms, in the logical language of your choice. You’d then need to show the relation of your intended (non-standard) models to other non-standard models and to discuss the order structure in precise (that is, mathematically rigorous) terms. And you’d need to prove theorems about the novelty of the structure, and whether its operations are Turing recursive or not. If not, what are they?
It looks like you’re trying to define a countable algebraic field which is linearly ordered and which has endpoints. Prove that it is not isomorphic to a subset of the rational numbers. And prove that it exists, at least by giving precise axioms and showing some relative consistency.

These infinite integers are set out in, e.g, Abraham Robinson’s Non-Standard Analysis. Or Goldblatt’s Lectures on the Hyperreals. (At the end of the day I do think that they are ‘really’ just large finite numbers. But not much hinges on that claim.) So they would seem to certainly exist, by whatever standard of existence one uses.
The basic idea is you forget everything about the infinite. Then agree on the finite numbers. Then ask, what are the infinite numbers? It just seems clear that infinite integers have the best claim to that title, in virtue of being so much like the finite numbers. The Cantorian infinite and systems like Conway’s don’t even give infinite integers much competition, as far as I can see.

I have no time to do any more than to quote from Wikipedia Criticism of Non-Standard Analysisâ article: âAbraham Robinson’s non-standard analysis does not need any axioms beyond ZermeloâFraenkel set theory(ZFC) (as shown explicitly by Wilhelmus Luxemburg’s ultrapower construction of the hyperreals), while its variant by Edward Nelson, known as IST, is similarly a conservative extension of ZFC.â Beyond Wikipedia, the relevant articles are easy to find.

Certainly, in ZFC there are hypernaturals. But whether they count as “numbers” is not clear. And likewise in ZFC there is a set of natural numbers. Which is crucially used in the construction of the hypernaturals.

Why wouldn’t they count as numbers?
“And likewise in ZFC there is a set of natural numbers. Which is crucially used in the construction of the hypernaturals.”
In reply, I note that atoms are used in the construction of ducks, but no one runs around demanding that atoms are ducks. Similarly, ‘the’ natural numbers are used in the construction of the infinite numbers (hypernaturals), but that doesn’t mean that the natural numbers are themselves a number.
I have a paper appearing soon in The Reasoner — two concepts of completing an infinite number of tasks. There, I discuss infinite tasks using hypernaturals — and then things are nicely like the finite case. Also, to repeat, when referring to the correct infinite numbers, paradoxes aren’t a dime a dozen.
Perhaps part of the issue is that I take my main point to be more of a conceptual one than a set theoretic one.

Also Eric — I was never claiming that I invented these numbers. The crucial point is the recognition that they ARE the infinite numbers. (And related points regarding the natural numbers I also think important.)

Jeremy:
Atoms aren’t ducks, but it’s tough to defend on an ontology on which there are ducks, and ducks are made of atoms, and there are no atoms. (Tough but maybe not impossible. For Aristotelian reasons, I am drawn to ontologies like that.)
If you do grant that there are sets like the set of natural numbers, then I don’t see how you get out of the paradoxes. Nothing in the paradoxes has much or anything to do with the claim that omega is a NUMBER.

Jeremy:
The posted statements about ZFC refute pretty much everything you’ve said so far, especially about how Aristotle is right and Cantor is wrong.
I eagerly welcome any new mathematics, especially the study of new order-types (and how they might relate to probability or computability). And yet, repeated asked to provide some actual mathematics, you tergiversate.
You’ve done no new math, discovered no new structure, proven no new theorem, and given not one single reason besides your own insistent yet merely verbal opinion to regard the objects you describe as numbers.
My conclusion, finally, is that you have no new math to offer, and not one single mathematical reason to think that you are talking about numbers of any kind.

Alex:
“Atoms aren’t ducks, but it’s tough to defend on an ontology on which there are ducks, and ducks are made of atoms, and there are no atoms. (Tough but maybe not impossible. For Aristotelian reasons, I am drawn to ontologies like that.)”
This is a good point. Ultimately I suppose that I think that there are things like natural number systems — but I haven’t worked much on the specifics here. I do want to go through The Foundations of Mathematics in the Theory of Sets a bit more carefully sometime.
“If you do grant that there are sets like the set of natural numbers, then I don’t see how you get out of the paradoxes. Nothing in the paradoxes has much or anything to do with the claim that omega is a NUMBER.”
Everything has to do with NUMBER! If whole numbers are dealt with in the finite case, then the only sensible numbers are the finite natural numbers. Similarly, if whole numbers (or the question: how many?) arises in the infinite case, the only sensible numbers (replies) are the infinite natural numbers. To the person who comes along and says “But I want to talk about countably many” I genuinely believe that the proper reply is (and at the risk of being a bit blunt): “But you are not talking about numbers. And you must be talking about numbers. How our concepts divide up the world is important, and not “merely verbal.” Furthermore, the structure you describe is potential and always becoming. Sane people have seen this since at least Aristotle. That anyone could think differently means that they are part of one of the silliest mass delusions in the history of ideas.”
And, I am willing to change this view. I will change it at the first person who explains to me what happens when two Zeno-spheres collide in a manner that does not make me want to tear my eyes out.

But prior to this discussion of bijections, a question arises: How should the relative sizes of infinite sets be determined? Bijection and subset are two possibilities. The current insistence on bijection is part of the mass confusion, I think. I also think that on the correct conception of infinite number, subset is the correct way to judge. The problem with bijections is that they ‘run out’. Here, I think that the internal/external distinction is trying to tell us something very important.

It’s certainly an interesting question how the relative sizes of infinite sets should be determined. But many of the paradoxes do not require this question to be answered because the concept of the size of a set does not need to enter into them.
Take Hilbert’s Hotel as the simplest case. To set it up, we need to suppose a hotel such that the set of its rooms is in bijection with the natural numbers. I said nothing about size (unless size is supposed to be understood via bijections).
Of course, size will enter in if you suppose that every set has a size. But why think that?

I think there are two replies here. One is, the paradoxes are telling us that the natural numbers are not a completed, actual infinity. And two, I think that people think that they are really talking about the single, correct conception of infinite number, whereas they, in fact, aren’t. So the question arises — why talk about bijections with the natural numbers at all? At the least, people shouldn’t think that they are investigating number, when addressing this question.

âAnd two, I think that people think that they are really talking about the single, correct conception of infinite number, whereas they, in fact, aren’t. So the question arises — why talk about bijections with the natural numbers at all? At the least, people shouldn’t think that they are investigating number, when addressing this question.â
I wanted to follow up a bit on this line of thought. I have been saying all along that I think that concepts matter, that what we call things matters. At the risk of giving a silly example, and one that I have given before, I think that the following comparison may be helpful. Imagine that people believe that the male birds are the male birds, but the female birds are cars. So when someone says âthere is a female birdâ, they are pointing at a car. If people genuinely carved up the world in this way, dissertations would be written on the bird-mating-problem â why canât male birds and âfemale birdsâ produce viable offspring? Now, this seems like a silly topic to write on. But if your concepts carved up the world in this way, it would be a problem. And it could be addressed in interesting ways (physical, biologicalâ¦). Of course, the correct reply, I think, is to point out â but these cars are not the female birds; your concepts are carving up the world in the wrong manner and are leading you into trouble. Similarly, the correct reply, I believe, to almost all problems involving infinite number, is to say, but that isnât infinite number that you are talking about; your concepts are carving up the world in the wrong way. The bird example likely seems silly to most people and the infinite number example less so, but I find them both equally silly. (This is the conceptual confusion and mass delusion that I referred to earlier; put simply, the Cantorian infinite does not hold a candle to infinite natural numbers for the claim to be the infinite natural numbers.) Once it becomes obvious to you what the infinite natural numbers are, the two examples are exactly on par. I would be interested if people thought that there were disanalogies between the bird and the infinite number case. I am having trouble seeing them, if there are.

Here’s one reason to talk about bijections. You think it’s possible to have a collection of rooms numbered by 1,…,M for some hyperreal infinity M. Now, there is an injection from the natural numbers to the rooms (e.g., defined inductively by f(1)=room 1, and f(n+1)=room with number one higher than f(n)). This comes up quite naturally–we’re counting those of the rooms that can be counted by finite numbers. It now seems not unnatural to ask: “Wouldn’t it be possible to have a hotel where one boards up all the rooms that are not in the range of this injection?” And plausibly the answer is affirmative. But once one boards up those rooms, one has a bijection between the natural numbers and the remaining rooms.
Moreover, many of the Hilbert’s Hotel puzzles come back even with the hyperreal infinite number M of rooms. It’s weird to suppose that every room is filled and yet there is room for more. But, of course we can. We suppose the rooms are numbered by 1,…,M, and then we instruct the person in room n to move to room n+1 if n is finite and to stay put if n is infinite. Of course, these instructions cannot be implemented as an internal function, but why does that matter?

If some whole number of rooms is boarded up, the number must be finite or infinite, that is, either a finite natural number or an infinite natural number.
âMoreover, many of the Hilbert’s Hotel puzzles come back even with the hyperreal infinite number M of rooms. It’s weird to suppose that every room is filled and yet there is room for more. But, of course we can. We suppose the rooms are numbered by 1,…,M, and then we instruct the person in room n to move to room n+1 if n is finite and to stay put if n is infinite. Of course, these instructions cannot be implemented as an internal function, but why does that matter?â
Because I think that the model is correct, and that M+1 really is one greater than M. And so, a guest cannot be added. Why not? Here I suppose some metaphysics does enter the picture. Infinite natural numbers are large finite numbers. Why? Well, because if a large finite number is investigated under reasonable assumptions â the structure turns out to that of an infinite natural number, and I donât think that that fact is merely coincidental. That argument is in two places:http://philpapers.org/rec/GWIINAhttp://philpapers.org/rec/GWIOIN
So the finite natural numbers become infinite natural numbers, when you consider the sequence in base 1: | || ||| |||| ||||| |||||| |||||||â¦
Thus, any sort of bijection eventually âfades outâ. And, a guest cannot be added to a full hotel with M guests, where M is infinite.

There is also a somewhat simpler reply. When one says something like — ‘board up the rooms with finite numbers’ — the only reason to think that this is possible is if one thinks that the natural numbers form an actual, determined infinity. But to think this is incorrect — because such an infinity is merely potential.
It’d be just the same as saying — ‘go paint a ray’ — where again the only reason to think that this is possible is if one thought that a ray was an actual, determined distance (infinity). But again, this strikes me as clearly wrong.